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HITs-Examples
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Switch to the DecidablePaths typeclass

This commit is contained in:
Dan Frumin 2017-05-23 12:12:22 +02:00
parent ebdaab4de0
commit 5abf0acf06
1 changed files with 7 additions and 13 deletions
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20
FinSets.v
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@ -331,9 +331,7 @@ End FinSet.
Section functions. Section functions.
Parameter A : Type. Parameter A : Type.
Context (A_eqdec: forall (x y : A), Decidable (x = y)). Context {A_eqdec: DecidablePaths A}.
Definition deceq (x y : A) :=
if dec (x = y) then true else false.
Notation "{| x |}" := (L x). Notation "{| x |}" := (L x).
Infix "∪" := U (at level 8, right associativity). Infix "∪" := U (at level 8, right associativity).
@ -358,7 +356,7 @@ Proof.
intros a. intros a.
hrecursion. hrecursion.
- exact false. - exact false.
- intro a'. apply (deceq a a'). - intro a'. destruct (dec (a = a')); [exact true | exact false].
- apply orb. - apply orb.
- intros x y z. compute. destruct x; reflexivity. - intros x y z. compute. destruct x; reflexivity.
- intros x y. compute. destruct x, y; reflexivity. - intros x y. compute. destruct x, y; reflexivity.
@ -457,7 +455,6 @@ hinduction.
all: try (intros; apply path_forall; intro; apply set_path2). all: try (intros; apply path_forall; intro; apply set_path2).
- intro Y. cbv. reflexivity. - intro Y. cbv. reflexivity.
- intros a Y. cbn. - intros a Y. cbn.
unfold deceq;
destruct (dec (a = a)); simpl. destruct (dec (a = a)); simpl.
+ reflexivity. + reflexivity.
+ contradiction n. reflexivity. + contradiction n. reflexivity.
@ -494,7 +491,7 @@ hrecursion X; try (intros; apply set_path2).
- cbn. unfold intersection. intros a. - cbn. unfold intersection. intros a.
hrecursion Y; try (intros; apply set_path2). hrecursion Y; try (intros; apply set_path2).
+ cbn. reflexivity. + cbn. reflexivity.
+ cbn. intros. unfold deceq. + cbn. intros.
destruct (dec (a0 = a)). destruct (dec (a0 = a)).
rewrite p. destruct (dec (a=a)). rewrite p. destruct (dec (a=a)).
reflexivity. reflexivity.
@ -504,17 +501,14 @@ hrecursion X; try (intros; apply set_path2).
contradiction n. apply p^. reflexivity. contradiction n. apply p^. reflexivity.
+ cbn -[isIn]. intros Y1 Y2 IH1 IH2. + cbn -[isIn]. intros Y1 Y2 IH1 IH2.
rewrite IH1. rewrite IH1.
rewrite IH2. rewrite IH2.
symmetry. apply (comprehension_union (L a)).
rewrite (comprehension_union (L a)).
reflexivity.
- intros X1 X2 IH1 IH2. - intros X1 X2 IH1 IH2.
cbn. cbn.
unfold intersection in *. unfold intersection in *.
rewrite <- IH1. rewrite <- IH1.
rewrite <- IH2. rewrite <- IH2. symmetry.
rewrite comprehension_union. apply comprehension_union.
reflexivity.
Defined. Defined.